Question: Simplify and expand the following expression: $ \dfrac{3y + 9}{y - 4}+\dfrac{2y + 2}{2y - 10} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(y - 4)(2y - 10)$ Multiply the first term by $\dfrac{2y - 10}{2y - 10}$ $ \begin{align*} \dfrac{3y + 9}{y - 4} \times \dfrac{2y - 10}{2y - 10} & = \dfrac{(3y + 9)(2y - 10)}{(y - 4)(2y - 10)} \\ & = \dfrac{6y^2 - 12y - 90}{(y - 4)(2y - 10)}\end{align*} $ Multiply the second term by $\dfrac{y - 4}{y - 4}$ $ \begin{align*} \dfrac{2y + 2}{2y - 10} \times \dfrac{y - 4}{y - 4} & = \dfrac{(2y + 2)(y - 4)}{(2y - 10)(y - 4)} \\ & = \dfrac{2y^2 - 6y - 8}{(2y - 10)(y - 4)}\end{align*} $ Now we have: $ = \dfrac{6y^2 - 12y - 90}{(y - 4)(2y - 10)} + \dfrac{2y^2 - 6y - 8}{(2y - 10)(y - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{6y^2 - 12y - 90 + 2y^2 - 6y - 8}{(y - 4)(2y - 10)} $ $ = \dfrac{8y^2 - 18y - 98}{(y - 4)(2y - 10)}$ Expand the denominator: $ = \dfrac{8y^2 - 18y - 98}{2y^2 - 18y + 40}$ Simplify: $ = \dfrac{4y^2 - 9y - 49}{y^2 - 9y + 20}$